Classifications

• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M4/36—Selection of substances as active materials, active masses, active liquids
  • H01M4/362—Composites
  • H01M4/366—Composites as layered products
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01B—NON-METALLIC ELEMENTS; COMPOUNDS THEREOF; METALLOIDS OR COMPOUNDS THEREOF NOT COVERED BY SUBCLASS C01C
  • C01B32/00—Carbon; Compounds thereof
  • C01B32/05—Preparation or purification of carbon not covered by groups C01B32/15, C01B32/20, C01B32/25, C01B32/30
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01B—NON-METALLIC ELEMENTS; COMPOUNDS THEREOF; METALLOIDS OR COMPOUNDS THEREOF NOT COVERED BY SUBCLASS C01C
  • C01B33/00—Silicon; Compounds thereof
  • C01B33/02—Silicon
  • C01B33/021—Preparation
  • C01B33/027—Preparation by decomposition or reduction of gaseous or vaporised silicon compounds other than silica or silica-containing material
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M10/00—Secondary cells; Manufacture thereof
  • H01M10/05—Accumulators with non-aqueous electrolyte
  • H01M10/052—Li-accumulators
  • H01M10/0525—Rocking-chair batteries, i.e. batteries with lithium insertion or intercalation in both electrodes; Lithium-ion batteries
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M4/36—Selection of substances as active materials, active masses, active liquids
  • H01M4/362—Composites
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M4/36—Selection of substances as active materials, active masses, active liquids
  • H01M4/362—Composites
  • H01M4/364—Composites as mixtures
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M4/36—Selection of substances as active materials, active masses, active liquids
  • H01M4/38—Selection of substances as active materials, active masses, active liquids of elements or alloys
  • H01M4/386—Silicon or alloys based on silicon
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M4/36—Selection of substances as active materials, active masses, active liquids
  • H01M4/58—Selection of substances as active materials, active masses, active liquids of inorganic compounds other than oxides or hydroxides, e.g. sulfides, selenides, tellurides, halogenides or LiCoFy; of polyanionic structures, e.g. phosphates, silicates or borates
  • H01M4/583—Carbonaceous material, e.g. graphite-intercalation compounds or CFx
  • H01M4/587—Carbonaceous material, e.g. graphite-intercalation compounds or CFx for inserting or intercalating light metals
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M4/62—Selection of inactive substances as ingredients for active masses, e.g. binders, fillers
  • H01M4/624—Electric conductive fillers
  • H01M4/625—Carbon or graphite
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01P—INDEXING SCHEME RELATING TO STRUCTURAL AND PHYSICAL ASPECTS OF SOLID INORGANIC COMPOUNDS
  • C01P2002/00—Crystal-structural characteristics
  • C01P2002/70—Crystal-structural characteristics defined by measured X-ray, neutron or electron diffraction data
  • C01P2002/74—Crystal-structural characteristics defined by measured X-ray, neutron or electron diffraction data by peak-intensities or a ratio thereof only
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01P—INDEXING SCHEME RELATING TO STRUCTURAL AND PHYSICAL ASPECTS OF SOLID INORGANIC COMPOUNDS
  • C01P2002/00—Crystal-structural characteristics
  • C01P2002/80—Crystal-structural characteristics defined by measured data other than those specified in group C01P2002/70
  • C01P2002/82—Crystal-structural characteristics defined by measured data other than those specified in group C01P2002/70 by IR- or Raman-data
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01P—INDEXING SCHEME RELATING TO STRUCTURAL AND PHYSICAL ASPECTS OF SOLID INORGANIC COMPOUNDS
  • C01P2002/00—Crystal-structural characteristics
  • C01P2002/80—Crystal-structural characteristics defined by measured data other than those specified in group C01P2002/70
  • C01P2002/85—Crystal-structural characteristics defined by measured data other than those specified in group C01P2002/70 by XPS, EDX or EDAX data
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01P—INDEXING SCHEME RELATING TO STRUCTURAL AND PHYSICAL ASPECTS OF SOLID INORGANIC COMPOUNDS
  • C01P2004/00—Particle morphology
  • C01P2004/01—Particle morphology depicted by an image
  • C01P2004/03—Particle morphology depicted by an image obtained by SEM
• • C—CHEMISTRY; METALLURGY
  • C01—INORGANIC CHEMISTRY
  • C01P—INDEXING SCHEME RELATING TO STRUCTURAL AND PHYSICAL ASPECTS OF SOLID INORGANIC COMPOUNDS
  • C01P2006/00—Physical properties of inorganic compounds
  • C01P2006/14—Pore volume
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M2004/021—Physical characteristics, e.g. porosity, surface area
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
  • H01M4/00—Electrodes
  • H01M4/02—Electrodes composed of, or comprising, active material
  • H01M2004/026—Electrodes composed of, or comprising, active material characterised by the polarity
  • H01M2004/027—Negative electrodes
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
  • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
  • Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
  • Y02E60/10—Energy storage using batteries

Definitions

• One embodiment of the present invention relates to composite particles containing silicon and carbon, a negative electrode mixture layer containing the composite particles, and a lithium ion secondary battery containing the negative electrode mixture layer.
• the lithium-ion secondary batteries used in IT devices such as smartphones and tablet PCs, vacuum cleaners, power tools, electric bicycles, drones, and automobiles require negative electrode active materials that have both high capacity and high output.
• Silicon theoretical specific capacity: 4200 mAh/g
• graphite theoretical specific capacity: 372 mAh/g
• lithium-ion secondary batteries One of the major features of lithium-ion secondary batteries is their high volumetric energy density.
• the positive electrode and the negative electrode must have high electrode density and high capacity. Therefore, the negative electrode material and the positive electrode material that constitute it are also required to have a high powder compaction density and a high specific capacity.
• electrode density refers to the apparent density of the electrode mixture layer.
• negative electrode material refers to a negative electrode active material.
• cathode material refers to the cathode active material, as well as the anode material.
• the positive electrode material and/or the negative electrode material may be referred to as “electrode material”.
• Patent Document 1 artificial graphite having a specific surface area (BET) of 0.1 to 1.2 m 2 /g as a negative electrode active material and artificial graphite larger than the artificial graphite
• BET specific surface area
• a negative electrode comprising one or more selected from the group consisting of natural graphite and softened carbon having specific surface areas is disclosed.
• Patent Document 2 discloses a metal ion battery electrode comprising an active layer in contact with a current collector, wherein the active layer is silicon, the formula SiO x (where 0 ⁇ a plurality of porous particles (i) comprising an electroactive material selected from silicon oxide, germanium, tin, aluminum, and mixtures thereof, represented by x ⁇ 1.5), and having a diameter of 0.5 to 40 ⁇ m and a porous particle (i) having an intraparticle porosity of less than 30 %, and one or more of graphite, soft carbon, and hard carbon, wherein the D50 particle size is a plurality of carbon particles (ii) ranging from 1 to 100 ⁇ m, said active layer comprising at least 50% by weight of said carbon particles (ii), said carbon relative to the D50 particle size of said porous particles (i) Electrodes for metal ion batteries are disclosed in which the ratio of D50 particle sizes of particles (ii) is in the range of 1.5-30
• Patent Document 3 discloses a particulate material in which a porous carbon framework has a bimodal or multimodal pore size distribution.
• the negative electrode disclosed in Patent Document 1 contains only graphite as an active material and does not contain silicon, only the capacity of graphite can be expected.
• the metal ion battery electrode disclosed in Patent Document 2 only the size ratio of the porous particles (i) containing an electroactive substance and the carbon particles (ii) is controlled, and the electrode density and capacity obtained by optimizing it are controlled. I can only hope.
• Patent Document 3 focuses on improving rate characteristics, and does not necessarily improve electrode density or specific capacitance.
• the silicon content of these materials is increased, the initial specific capacity of the battery is high, but the expansion and contraction during charging and discharging are large, so it is considered that the electrode deteriorates severely and the cycle characteristics deteriorate.
• the problem to be solved by the present invention is to provide a negative electrode material for lithium ion secondary batteries that has a high compression density, a high silicon concentration, and good cycle characteristics.
• the present inventors have made intensive studies, and as a result, in composite particles containing silicon and carbon, the ratio of composite particles with a specific particle size to the whole, the average silicon content, and the composite The inventors have found that the above problems can be satisfactorily solved by setting the silicon content of the whole body particles to a specific range, and have completed the present invention.
• a configuration example of an embodiment of the present invention is as follows.
• Composite particles containing silicon and carbon When the cross-sectional diameter and silicon content of the composite particles are measured by cross-sectional SEM-EDS, and the composite particles having a cross-sectional diameter of 1/2 or less of the number average of the cross-sectional diameter are defined as small-sized composite particles, The ratio of the number of the small-diameter composite particles to the number of the measured composite particles is 5% or more and 50% or less, The ratio of the average silicon content (mass%) of the composite particles other than the small-diameter composite particles to the average silicon content (mass%) of the small-diameter composite particles is 0.90 or less, The silicon content of the entire composite particles is 45% by mass or more, composite particles.
• a porous carbon (S) having a D V50 of 1.0 to 10.0 ⁇ m and a pore volume of 0.8 to 2.2 cm 3 /g is allowed to act with a silane gas, and the porous carbon (S) ) 5 to 50% by mass of composite particles (S) obtained by filling the pores of silicon,
• a porous carbon (L) having a D V50 larger than that of the porous carbon (S) and a pore volume of 0.2 to 0.8 cm 3 /g is reacted with a silane gas to remove the porous carbon (L). 50 to 95% by mass of composite particles (L) obtained by filling silicon in the pores; (the total of the composite particles (S) and the composite particles (L) is 100% by mass), the composite particle according to any one of [1] to [7].
• a lithium ion secondary battery comprising the negative electrode mixture layer of [9].
• the composite particles of one embodiment of the present invention have a high compression density and a high specific capacity, and therefore can provide a negative electrode mixture layer and a lithium ion secondary battery with high volumetric energy density and good cycle characteristics.
• the composite particle of one embodiment of the present invention is a composite particle containing silicon and carbon.
• the carbon is not particularly limited, it preferably contains a porous carbon material. Since the carbon contains a porous carbon material, it is possible to absorb the volume change due to the expansion/contraction of silicon accompanying lithiation/delithiation. It is more preferable that the silicon be arranged in the pores of the porous carbon material, but the silicon may be present on the surface of the particles of the porous carbon material.
• the composite particles of one embodiment of the invention are controlled by cross-sectional diameter and silicon content as measured by cross-sectional SEM-EDS.
• cross-sectional SEM-EDS is performed on a composite particle group sampled from a homogeneous state. For example, a well-mixed powder of a large amount of composite particles is scooped with a microspatula or the like, placed on a carbon tape, and a cross-section is taken out with a cross-section polisher, etc., or the well-mixed powder is applied to the resin. Embedding and grinding this to generate a cross section can be mentioned.
• the cross-sectional diameter and silicon content of the composite particles can be determined by SEM-EDS of the cross section. can be examined. However, particles broken during electrode pressing are excluded from the measurement object. Whether cracking occurred during electrode pressing is readily discernible from adjacent composite particle shapes.
• cross-sectional diameter in the cross-sectional SEM image of the composite particles is the diameter of the cross-section of individual particles obtained by cross-sectional SEM observation of the composite particles. Equivalent circle diameter calculated from the cross-sectional area of
• the number average of the measured cross-sectional diameters of each composite particle is taken as the average cross-sectional diameter of the composite particles. That is, the average cross-sectional diameter is a value obtained by dividing the sum of cross-sectional diameter values obtained by the measured number of composite particles.
• Composite particles having a cross-sectional diameter of 1/2 or less of the average cross-sectional diameter will be referred to as "small-diameter composite particles".
• Composite particles other than the small-diameter composite particles are called "large-diameter composite particles".
• the ratio of the number of small-diameter composite particles to the measured number of composite particles is 5% or more.
• the composite particles having a size similar to that of the small-diameter composite particles can efficiently enter the gaps of the other particles, and the powder formed by the present composite particles can be Can increase compaction density. From this point of view, the ratio is preferably 10% or more, more preferably 20% or more.
• the ratio of the number of the small-diameter composite particles to the total number of measured composite particles is 50% or less.
• the ratio is 50% or less, the present composite particles have few voids between particles having a size similar to that of the small-diameter composite particles, so that the compression density of the powder can be increased. From this point of view, the ratio is preferably 45% or less, more preferably 35% or less.
• the cross-sectional SEM image of the composite particles whose cross-sectional diameter was measured was analyzed using EDS for silicon and carbon in the center of the particles.
• the proportion of the element is calculated in units of % by mass.
• the numerical value (number average) obtained by averaging the silicon contents obtained from the measured individual composite particles using the number of particles is defined as the average silicon content of the composite particles. That is, the average silicon content is a value obtained by dividing the total silicon content of individual composite particles by the number of measured composite particles.
• the numerical value obtained by number-averaging only the small-diameter composite particles by the above method is called “average silicon content (x) of the small-diameter composite particles”. Further, a numerical value obtained by averaging only the large particle size composite particles is called “average silicon content (y) of the large particle size composite particles”.
• the value of (y)/(x) is 0.90 or less. That is, in the composite particles of one embodiment of the present invention, the silicon content of the small-diameter composite particles is higher than that of the large-diameter composite particles. This is a requirement based on the fact that composite particles with a small particle diameter are considered to be more resistant to cracking due to expansion and contraction than composite particles with a large particle diameter. As a result, the silicon content of the composite particles as a whole can be made higher than when all the composite particles have the same silicon content, so that the specific capacity of the composite particles can be increased.
• the value of (y)/(x) is preferably 0.85 or less, more preferably 0.80 or less.
• the silicon content of the composite particles of one embodiment of the present invention is 45% by mass or more.
• the composite particles can have a sufficiently high specific capacity. From this point of view, the silicon content is preferably 48% by mass or more, more preferably 50% by mass or more.
• the silicon content is preferably 85% by mass or less. When the silicon content is 85% by mass or less, good cycle characteristics are obtained in the battery. From this point of view, the silicon content is more preferably 75% by mass or less, and even more preferably 65% by mass or less.
• the content of silicon in the composite particles can be quantitatively analyzed by, for example, X-ray fluorescence analysis (XRF) or inductively coupled plasma emission spectroscopy (ICP-AES). This silicon content is sometimes specifically referred to as a "quantitative silicon content”.
• XRF X-ray fluorescence analysis
• ICP-AES inductively coupled plasma emission spectroscopy
• the pore volume of the composite particles of one embodiment of the present invention is preferably 0.10 cm 3 /g or less.
• the pore volume is 0.10 cm 3 /g or less, the specific surface area becomes small and decomposition of the electrolytic solution occurs only to an allowable level, so that the initial coulombic efficiency in the battery is excellent.
• the pore volume of 0.10 cm 3 /g or less reduces the chance of silicon coming into contact with oxygen and moisture in the air. less prone to oxidation. From this point of view, the pore volume is more preferably 0.05 cm 3 /g or less.
• the ratio of oxygen content to silicon content is preferably 0.001 or more.
• the unit of oxygen content and silicon content is % by mass.
• the ratio of the oxygen content to the silicon content is more preferably 0.002 or more, and even more preferably 0.005 or more.
• the ratio of oxygen content to silicon content is preferably 0.300 or less. If the ratio is 0.300 or less, the irreversible capacity due to silicon oxide is small. From this point of view, the ratio of the oxygen content to the silicon content is more preferably 0.200 or less, and even more preferably 0.100 or less.
• the oxygen content in the composite particles can be measured, for example, by an oxygen analyzer that heats a sample together with carbon at a high temperature in an inert gas and quantifies the generated CO and CO 2 with an infrared detector.
• an oxygen analyzer that heats a sample together with carbon at a high temperature in an inert gas and quantifies the generated CO and CO 2 with an infrared detector.
• the content of silicon can be measured by the above XRF or ICP-AES.
• the composite particles of one embodiment of the present invention preferably have a peak in the range of 450 to 495 cm ⁇ 1 in Raman spectrum. The presence of a peak in this range suggests that the composite particles contain amorphous silicon. More preferably, the composite particles of one embodiment of the present invention have a peak in the range of 450 to 495 cm -1 and not in the range of 500 to 530 cm -1 in the Raman spectrum. The absence of a peak in the 500-530 cm -1 range suggests that the composite particles do not contain crystalline silicon. Since amorphous silicon expands and contracts more isotropically than crystalline silicon during lithiation, it is thought that composite particles and electrode mixture layers that are more resistant to mechanical deterioration can be obtained.
• the composite particles of one embodiment of the present invention have an XRD pattern obtained by powder X-ray diffraction measurement (powder XRD measurement) using Cu—K ⁇ rays, in which (the peak height of the 111 plane of SiC) and (the 111 plane of Si is preferably 0.010 or less.
• the ratio is 0.010 or less, the composite particles contain less silicon carbide (SiC), resulting in a high specific capacity. From this point of view, the ratio is more preferably 0.005 or less, more preferably 0.001 or less, and most preferably 0.000.
• the ratio is also expressed as (I SiC111 )/(I Si111 ).
• the composite particle of one embodiment of the present invention may have a coating layer on the particle surface.
• Specific coats include carbon coats, inorganic oxide coats, and polymer coats.
• Methods of carbon coating include chemical vapor deposition (CVD) and physical vapor deposition (PVD).
• Methods of inorganic oxide coating include CVD, PVD, Atomic Layer Deposition (ALD), wet methods, and the like.
• the wet method includes a method in which the composite particles are coated with a liquid obtained by dissolving or dispersing an inorganic oxide precursor (metal carboxylate or alkoxide) in a solvent, and the solvent is removed by heat treatment or the like.
• Examples of the type of polymer coating include a method of coating using a polymer solution, a method of coating using a polymer precursor containing a monomer, and polymerizing by the action of temperature, light, etc., or a combination thereof. .
• the inorganic oxide is selected from the group consisting of oxides of Al, Ti, V, Cr, Mn, Fe, Co, Ni, Y, Zr, Mo, Nb, La, Ce, Ta, W and Li-containing oxides One or more are preferred.
• the coat layer may be used alone, or may be a combination of multiple types. If heating is required during coating, it is preferably done at less than 800° C. to avoid silicon in the composite particles reacting with carbon to form silicon carbide.
• composite particles (S) and composite particles (L) in which silicon is filled in porous carbon are produced in the method for producing composite particles described later (step After 1 or 2), the composite particles (S), the composite particles (L), or both the composite particles (S) and the composite particles (L) need to be subjected to the above treatment.
• the composite particles (S) and the composite particles (L) are each coated and then mixed (step 3). or coated after mixing (step 3).
• Effects of the coat layer include, for example, suppression of oxidation of silicon in the composite particles over time, increase in initial coulomb efficiency, and improvement in cycle characteristics, as shown below.
• silicon is oxidized over time.
• the existence of the coat layer on the surface of the composite particle can suppress the entry of air or oxygen-containing gas into the interior of the composite particle.
• the electrolytic solution decomposition product film (SEI ⁇ Solid Electrolyte Interface>
• the initial coulombic efficiency decreases due to the presence of lithium ions that cannot escape from the closed pores in the composite particles.
• the SEI film is present, so the proportion of lithium ions trapped in the composite particles is greatly reduced.
• the coating layer is present on the surface of the composite particles, the insertion of lithium ions into the pores that are likely to be blocked by the SEI coating is prevented, thereby increasing the initial coulombic efficiency.
• silicon in the composite particles is thought to react with fluorine, which is a component element of the electrolyte, and dissolve out as a silicon fluoride compound. Elution of silicon reduces the specific capacity of the composite particles.
• the coating layer is present on the composite particle surface, the elution of the Si-containing compound is suppressed, the decrease in the specific capacity of the composite particle is suppressed, and the side reaction with the electrolyte is also suppressed.
• the coat layer reduces the resistance and improves the rate characteristics.
• the present composite particles When the present composite particles have a coat layer, they may have the same coat layer, but the presence or absence of the coat layer and the type thereof may differ depending on the particle diameter. For example, there are the following combinations. (i) A mode in which the small-diameter composite particles have a coat layer and the large-diameter composite particles do not have a coat layer. (ii) A mode in which the small-diameter composite particles do not have a coat layer, and the large-diameter composite particles have a coat layer. (iii) A mode in which the small-diameter composite particles and the large-diameter composite particles have coat layers, but the types of the respective coat layers are different.
• the surface coating of composite particles can be analyzed by analyzing the particle surface. Examples thereof include SEM-EDS, Auger electron spectroscopy, microscopic infrared spectroscopy, and microscopic Raman spectroscopy. Since the surface of the composite particles is analyzed, the spatial resolution of the analyzer is preferably 1/2 or less of the average particle diameter of the composite particles.
• the small-diameter composite particles have a high silicon content, it is preferable that the small-diameter composite particles have a coating layer.
• the thickness of the coating layer is not limited as long as it does not impair the battery performance.
• the thickness of the coat layer can be measured by a transmission electron microscope or a cross-sectional SEM. Alternatively, it may be a value calculated from the content of elements derived from the coating layer quantified by elemental analysis of the composite particles and the sphere-equivalent surface area obtained from the average cross-sectional diameter of the composite particles by SEM as described above. The calculation method will be explained in detail. For example, the calculation of the coat thickness of the small-diameter composite particles when the coat is confirmed on the small-diameter composite particles by SEM-EDS is performed by the following method.
• the small particle size composite particles can be calculated.
• the ratio of the small-diameter composite particles to the coat component can be calculated from the content of elements derived from the coat layer quantified by elemental analysis of the composite particles and the mass concentration thereof. From the average cross-sectional diameter of the small particle size composite particles in the composite particles, the sphere equivalent surface area can be obtained, and from the average composition of the small particle size composites, the average mass of the small particle size composites can also be calculated.
• the mass of the element derived from the coat layer present on the surface of the small particle size composite particles is calculated.
• the thickness of the coating layer can be calculated using this value, the surface area equivalent to a sphere, and the density of the type of coating layer obtained previously.
• the densities of the coating layer species are appropriately selected from literature values and used.
• the fact that the elements are detected from the small-diameter composite particles means that the surfaces of the small-diameter composite particles are coated with oxides of these elements. Cycle characteristics can be improved by having a coat layer of oxides of one or more of these elements on the surface of the composite particles.
• the method for producing composite particles according to an embodiment of the present invention is not particularly limited, but examples include a production method including the following steps.
• Step 1 A porous carbon (S) having a D V50 of 1.0 to 10.0 ⁇ m and a pore volume of 0.8 to 2.2 cm 3 /g is reacted with a silane gas to obtain the porous carbon (S ) to produce composite particles (S) in which silicon is filled in the pores.
• Step 2 A porous carbon (L) having a D V50 greater than that of the porous carbon (S) and a pore volume of 0.2 to 0.8 cm 3 /g is allowed to act with silane gas to fill the pores with silicon.
• Step 3 Mix the composite particles (S) and the composite particles (L) so that the ratio of the composite particles (S) is 5 to 50% by mass (composite particles (S) and composite particles (L) is 100% by mass) step.
• D V50 is the 50% particle size in the volume-based cumulative particle size distribution. It can be measured using a particle size distribution meter using a laser diffraction method.
• a porous carbon (S) having a D V50 of 1.0 to 10.0 ⁇ m, such as 1.0 to 4.0 ⁇ m, and a pore volume of 0.8 to 2.2 cm 3 /g is supplied with silane gas. act. This results in composite particles with a relatively low D V50 and a relatively high silicon content.
• the porous carbon (S) is placed in a tubular furnace and heated to 350° C. to 450° C. in an inert atmosphere such as argon or nitrogen gas.
• SiH 4 silane
• the concentration of silane in the reaction gas may be 100% by volume, but it is also possible to use, for example, a mixture of nitrogen gas and argon gas with a concentration of 0.01 to 99.9% by volume.
• the pressure during the reaction may be normal pressure (101 ⁇ 10 kPaA).
• the silane gas is adsorbed in the pores in the porous carbon (S) and further thermally decomposed.
• silicon is deposited in the pores of the porous carbon (S).
• Silane gas is converted into silicon and hydrogen by the reaction shown in the following formula.
• the gas is switched from silane to an inert gas.
• a post-process when the temperature is lowered to 100 to 50° C. under an inert gas atmosphere, it is exposed to air or oxygen gas to form an oxide film on the silicon surface.
• the gas may be switched from silane to a hydrocarbon gas to form a carbon film on the surface of the composite particles by CVD.
• These post-processes can be performed alone or in combination. When used in combination, either can be performed first.
• the Si--C composites obtained in step 1 are called "composite particles (S)".
• Step 2 the porous carbon (L) having a D V50 greater than that of the porous carbon (L) and a pore volume of 0.2 to 0.8 cm 3 /g is reacted with silane gas.
• D V50 of porous carbon (S) is 4.0 ⁇ m or less
• porous carbon (L) with D V50 greater than 4.0 ⁇ m is used. This results in composite particles with relatively high D V50 and relatively low silicon content.
• the reaction conditions are the same as in step 1, and the only difference is the raw material porous carbon.
• the Si—C composites obtained in step 2 are called “composite particles (L)”.
• Step 3 This is a step of preparing the composite particles according to the present invention by mixing the composite particles (S) and the composite particles (L) in an appropriate ratio.
• the composite particles (S) are added so as to be 5 to 50% by mass of the whole, and the remainder is the composite particles (L). That is, the ratio of the composite particles (S) to the total of the composite particles (S) and the composite particles (L) is 5 to 50% by mass, and the ratio of the composite particles (L) is 50 to 95% by mass. is.
• the "small-diameter composite particles” do not necessarily consist entirely of the composite particles (S). Also, the “large particle size composite particles” are not necessarily composed entirely of the composite particles (L).
• the ratio of the desired small-diameter composite particles, the average cross-sectional diameter, the average silicon content, (the ratio of the large-diameter composite particles In order to produce the average silicon content (y))/(average silicon content (x) of the small particle size composite particles) and the silicon content by quantitative analysis for the entire composite particles, the composite It is necessary to analyze the composite particles obtained by blending the particles (S) and the composite particles (L) by cross-sectional SEM-EDS, and to make it while confirming that the desired product is obtained.
• three or more types of composite particles can be blended.
• the porous carbon (L) of 2 to 0.8 cm 3 /g commercially available activated carbon or carbon molecular sieves may be purchased and used. You may synthesize
• the DV50 can be adjusted by appropriately performing pulverization, sieving, air classification, etc. on the purchased material, or on the material before, during, or after the manufacturing process of the porous carbon.
• the boundary value of D V50 of composite particles (S) and composite particles (L) is arbitrary and can be any value between 1 and 10 ⁇ m, for example 4 ⁇ m.
• the D V50 of the composite particles (S) may be 4 ⁇ m or less
• the D V50 of the composite particles (L) may be greater than 4 ⁇ m
• the D V50 of the composite particles (S) may be 10 ⁇ m or less.
• D V50 of (L) may be greater than 10 ⁇ m.
• the composite particles of one embodiment of the present invention can be widely used as an electrode material for metal ion secondary batteries, and particularly a negative electrode material for lithium ion secondary batteries. can be preferably used as.
• the composite particles may be used alone, but for example, for the purpose of adjusting the battery capacity, or for the purpose of absorbing volume changes due to expansion and contraction of the composite particles, other negative electrode materials are used together. good too.
• other negative electrode materials those commonly used in lithium ion secondary batteries can be used.
• the composite particles are usually mixed with the other negative electrode material.
• Examples of other negative electrode materials include graphite, hard carbon, soft carbon, lithium titanate (Li 4 Ti 5 O 12 ), alloy active materials such as silicon and tin, and composite materials thereof. These negative electrode materials are usually used in the form of particles. As the negative electrode material other than the composite particles, one kind or two or more kinds may be used. Among them, graphite and hard carbon are particularly preferably used.
• the form of the negative electrode material of the present invention containing composite particles and graphite particles is one of the preferred forms from the viewpoint of adjusting the capacity and reducing the volume change of the entire negative electrode mixture layer. When a plurality of types of materials are used as the negative electrode material, they may be used after being mixed in advance, or may be sequentially added when preparing a slurry for forming a negative electrode mixture, which will be described later.
• a commercially available mixer or stirrer can be used as a device for mixing the composite particles and other materials.
• Specific examples include mixers such as a mortar, a ribbon mixer, a V-type mixer, a W-type mixer, a one-blade mixer, and a Nauta mixer.
• the negative electrode mixture layer of one embodiment of the present invention contains the negative electrode material. That is, the negative electrode mixture layer of one embodiment of the present invention contains the present composite particles.
• the negative electrode mixture layer of one embodiment of the present invention can be used as a negative electrode mixture layer for a lithium ion secondary battery.
• the negative electrode mixture layer is generally composed of a negative electrode material, a binder, and a conductive aid as an optional component.
• a slurry for forming a negative electrode mixture layer is prepared using a negative electrode material, a binder, a conductive additive as an optional component, and a solvent.
• the slurry is applied to a current collector such as copper foil and dried. It is further vacuum dried to remove the solvent.
• the obtained product is sometimes called a negative electrode sheet.
• the negative electrode sheet consists of a negative electrode mixture layer and a current collector.
• the negative electrode sheet is cut or punched into a required shape and size, and then pressed to increase the density of the electrode mixture layer (sometimes referred to as electrode density). Improving the electrode density improves the energy density of the battery.
• the electrode density varies depending on the components of the electrode mixture layer, but is 1.1 to 1.9 g/cm 3 .
• the pressing method is not particularly limited as long as the desired electrode density can be achieved, but uniaxial pressing, roll pressing, and the like can be mentioned.
• the process is exemplified by performing pressing after shape processing, but shape processing may be performed after pressing.
• the object in this state is called a negative electrode in the present invention.
• the negative electrode also includes a state in which a current collector is attached with a current collecting tab, if necessary.
• any binder commonly used in the negative electrode mixture layer of lithium ion secondary batteries can be freely selected and used.
• examples include polyethylene, polypropylene, ethylene propylene terpolymer, butadiene rubber, styrene butadiene rubber, butyl rubber, acrylic rubber, polyvinylidene fluoride, polytetrafluoroethylene, polyethylene oxide, polyepichlorohydrin, polyphosphazene, polyacrylonitrile, carboxymethylcellulose and its salts, polyacrylic acid, polyacrylamide, and the like.
• a binder may be used individually by 1 type, or may use 2 or more types.
• the amount of the binder is preferably 0.5 to 30 parts by mass with respect to 100 parts by mass of the negative electrode material.
• the conductive aid is not particularly limited as long as it serves to impart conductivity and dimensional stability (action to absorb volume changes of the composite particles during lithium insertion/extraction) to the electrode.
• carbon nanotubes, carbon nanofibers, vapor grown carbon fibers eg, "VGCF (registered trademark)-H” manufactured by Showa Denko KK
• conductive carbon black eg, "Denka Black (registered trademark)” Denka Co., Ltd., "SUPER C65” Imerys Graphite & Carbon, “SUPER C45” Imerys Graphite & Carbon
• conductive graphite for example, "KS6L” Imerys Graphite & Carbon, “SFG6L” Imerys ⁇ Graphite & Carbon Co., Ltd.
• two or more kinds of the conductive aids can be used.
• Carbon nanotubes, carbon nanofibers and vapor-grown carbon fibers are preferably included, and the fiber length of these conductive aids is preferably 1/2 or more of the DV50 of the composite particles. With this fiber length, these conductive aids bridge between the negative electrode active materials containing the composite particles, and the cycle characteristics can be improved.
• the fiber length of these conductive aids bridge between the negative electrode active materials containing the composite particles, and the cycle characteristics can be improved.
• the number of cross-links increases with the same amount of addition as compared to the case of using thicker ones. .
• it is more flexible it is more preferable from the viewpoint of improving the electrode density.
• the amount of the conductive aid is preferably 1 to 30 parts by mass with respect to 100 parts by mass of the negative electrode material.
• Solvents used in preparing slurry for electrode coating are not particularly limited, and include N-methyl-2-pyrrolidone, dimethylformamide, isopropanol, water and the like. In the case of a binder using water as a solvent, it is also preferable to use a thickener together. The amount of the solvent is adjusted so that the slurry has a viscosity that facilitates coating on the current collector.
• a lithium ion secondary battery according to the present invention includes the negative electrode mixture layer.
• the lithium ion secondary battery is generally composed of a negative electrode composed of the negative electrode mixture layer and the current collector, a positive electrode composed of the positive electrode mixture layer and the current collector, and a non-aqueous electrolyte and a non-aqueous polymer electrolyte present therebetween. It includes at least one, a separator, and a battery case that houses them.
• the lithium ion secondary battery only needs to include the negative electrode material mixture layer, and other configurations including conventionally known configurations can be employed without particular limitations.
• the positive electrode mixture layer usually consists of a positive electrode material, a conductive aid, and a binder.
• a positive electrode in the lithium ion secondary battery may have a general configuration in a normal lithium ion secondary battery.
• the positive electrode material is not particularly limited as long as it can be electrochemically intercalated and deintercalated with lithium and the redox potential of these reactions is sufficiently higher than the redox potential of the negative electrode reaction.
• LiCoO 2 , LiNiO 2 , LiMn 2 O 4 , LiCo 1/3 Mn 1/3 Ni 1/3 O 2 , carbon-coated LiFePO 4 and mixtures thereof can be suitably used.
• the conductive aid, the binder, and the solvent for slurry preparation those mentioned in the section of the negative electrode can be used.
• Aluminum foil is preferably used as the current collector.
• non-aqueous electrolyte and the non-aqueous polymer electrolyte used in lithium ion batteries those known for lithium ion secondary batteries can be used.
• the nonaqueous electrolytic solution for example, a solution obtained by dissolving a lithium salt such as LiClO 4 , LiPF 6 , LiAsF 6 , LiBF 4 , LiSO 3 CF 3 , CH 3 SO 3 Li in the following solvent or polymer is used.
• Solvents include non-aqueous solvents such as ethylene carbonate, diethyl carbonate, dimethyl carbonate, ethyl methyl carbonate, propylene carbonate, butylene carbonate, acetonitrile, propionitrile, dimethoxyethane, tetrahydrofuran, and ⁇ -butyrolactone.
• non-aqueous solvents such as ethylene carbonate, diethyl carbonate, dimethyl carbonate, ethyl methyl carbonate, propylene carbonate, butylene carbonate, acetonitrile, propionitrile, dimethoxyethane, tetrahydrofuran, and ⁇ -butyrolactone.
• Non-aqueous polymer electrolytes include, for example, gel polymer electrolytes containing polyethylene oxide, polyacrylonitrile, polyvinylidene fluoride, polymethyl methacrylate, and the like; solid polymer electrolytes containing polymers having ethylene oxide bonds, and the like. be done.
• a small amount of additives used in electrolyte solutions for general lithium-ion batteries may be added to the non-aqueous electrolyte solution.
• examples of such substances include vinylene carbonate (VC), biphenyl, propane sultone (PS), fluoroethylene carbonate (FEC), ethylene sultone (ES) and the like.
• VC and FEC are preferred.
• the amount to be added is preferably 0.01 to 20% by mass with respect to 100% by mass of the non-aqueous electrolytic solution.
• the separator can be freely selected from materials that can be used in general lithium-ion secondary batteries, including combinations thereof, and examples thereof include microporous films made of polyethylene or polypropylene. Moreover, it is also possible to use such a separator mixed with particles such as SiO 2 or Al 2 O 3 as a filler, or a separator adhered to the surface.
• the battery case is not particularly limited as long as it can accommodate the positive electrode, the negative electrode, the separator, and the electrolytic solution.
• those standardized in the industry such as commercially available battery packs, 18650-type cylindrical cells, coin-type cells, etc., they can be freely designed and used, such as those packed in aluminum packaging. can.
• the lithium ion secondary battery according to the present invention is a power source for electronic devices such as smartphones, tablet PCs, and personal digital assistants; a power source for electric motors such as electric tools, vacuum cleaners, electric bicycles, drones, and electric vehicles; It can be used for storage of electric power obtained by power generation, wind power generation, and the like.
• Apparatus NOVA4200e (registered trademark) manufactured by Quantachrome Instruments ⁇ Measurement gas: Nitrogen ⁇ Set value of relative pressure in the measurement range: 0.005 to 0.995
• the pore volume was obtained by calculating the adsorption amount at a relative pressure of 0.99 by linear approximation from the adsorption isotherm data at two points around the relative pressure of 0.99. At this time, calculations were made with a liquid nitrogen density of 0.808 g/cm 3 , a volume of 1 mol of nitrogen in the standard state of 22.4133 L, and an atomic weight of nitrogen of 14.0067.
• the particle diameter was obtained by measuring the length from the cross-sectional SEM image. In the case of spherical particles, the maximum diameter was taken as the particle diameter. In the case of non-spherical particles, the equivalent circle diameter was calculated from the cross-sectional area to obtain the particle diameter. The cross-sectional area was calculated using image analysis software (ImageJ). The method for measuring the cross-sectional diameter and the method for classifying composite particles into small particle size composite particles and large particle size composite particles are as described in the section of the mode for carrying out the invention.
• Si and C (carbon) content by EDS Si and C (carbon) content by EDS
• EDS XFlash (registered trademark) 5060 FlatQUAD
• the Si and C in the center of each particle of the cross-sectional sample were analyzed at an acceleration voltage of 5 kV, and from the ratios of these, the Si content (mass%) and C content were obtained. (% by mass) was calculated.
• the method for determining the average silicon (Si) content is as described in the section for carrying out the invention.
• Oxygen content About 20 mg of the sample was weighed into a nickel capsule, and the oxygen content was measured by the following method. ⁇ Apparatus: Oxygen/nitrogen analyzer EMGA-920 manufactured by Horiba, Ltd. ⁇ Carrier gas: Argon
• XRD device SmartLab (registered trademark) manufactured by Rigaku Corporation
• X-ray source Cu-K ⁇ ray K ⁇ ray elimination method: Ni filter X-ray output: 45 kV, 200 mA
• Measurement range 10.0-80.0°
• Scan speed 10.0°/min
• SBR Negative Electrode Sheet Styrene-butadiene rubber
• CMC carboxymethylcellulose
• Carbon black (SUPER C45, manufactured by Imerys Graphite & Carbon Co., Ltd.) and vapor-grown carbon fiber (VGCF (registered trademark)-H, manufactured by Showa Denko Co., Ltd.) were used as mixed conductive aids at a mass ratio of 3:2. A mixture was prepared.
• negative electrode material mixture of composite particles and human layer graphite
• 5 parts by mass of mixed conductive aid 2.5 parts by mass of CMC solid content, 2.5 parts by mass of SBR solid content
• a negative electrode material, a mixed conductive agent, a CMC aqueous solution and an SBR aqueous dispersion are mixed so as to become parts by mass, an appropriate amount of water for viscosity adjustment is added, and a rotation / revolution mixer (manufactured by Thinky Co., Ltd.) is used.
• the mixture was kneaded to obtain a slurry for forming a negative electrode mixture layer.
• the slurry concentration was 45-55 mass %.
• the slurry for forming the negative electrode mixture layer was uniformly applied onto a copper foil having a thickness of 20 ⁇ m as a current collecting foil using a doctor blade with a gap of 150 ⁇ m, dried on a hot plate, and then vacuum-dried at 70° C. for 12 hours. , a negative electrode mixture layer was formed on the current collector foil.
• a negative electrode sheet a sheet made up of a negative electrode mixture layer and a collector foil.
• the electrode density of the negative electrode was calculated as follows. The mass and thickness of the negative electrode obtained by the method described above were measured. The mass and thickness of the separately measured 16 mm diameter current collector foil were subtracted from the obtained values to obtain the mass and thickness of the negative electrode mixture layer, and the electrode density (negative electrode density) was calculated from these values.
• the electrolyte in the lithium counter electrode cell was 100 parts by mass of a solvent in which ethylene carbonate, ethylmethyl carbonate, and diethyl carbonate were mixed in a volume ratio of 3:5:2, 1 part by mass of vinylene carbonate (VC), and 1 part by mass of fluorocarbon.
• a liquid obtained by mixing 10 parts by mass of ethylene carbonate (FEC) and further dissolving the electrolyte lithium hexafluorophosphate (LiPF 6 ) in the mixture to a concentration of 1 mol/L was prepared.
• the specific capacity is a value obtained by dividing the capacity by the mass of the negative electrode active material.
• the "current value equivalent to 1C” is the mass of Si and carbon (including graphite) of the negative electrode active material contained in the negative electrode, and the theoretical specific capacity (4200 mAh / g and 372 mAh / g, respectively). It is the amount of current that can discharge the estimated capacity of the negative electrode in one hour.
• Three-electrode laminate type half cell [3-1] Preparation of three-electrode laminate type half cell [2-1] The negative electrode sheet obtained in [2-1] was rolled to a negative electrode mixture layer density of 1.3 to 1. 0.6 g/cm 3 , the area of the negative electrode mixture layer coated portion is 4.0 cm 2 (2.0 cm ⁇ 2.0 cm), and the negative electrode mixture layer uncoated portion (tab portion) is 0.5 cm. 2 (1.0 cm ⁇ 0.5 cm) was punched out to obtain a working electrode (negative electrode piece for working electrode).
• the Li roll was cut out to obtain a counter electrode Li piece with an area of 7.5 cm 2 (3.0 cm ⁇ 2.5 cm) and a reference electrode Li piece with an area of 3.75 cm 2 (1.5 cm ⁇ 2.5 cm). .
• a Ni tab with a width of 5 mm was prepared for a counter electrode and a reference electrode, and a Ni mesh of 5 mm ⁇ 20 mm was attached so as to overlap a 5 mm portion of the tip. At this time, the 5 mm width of the Ni tab and the 5 mm width of the Ni mesh were aligned and attached.
• a Ni tab for the working electrode was also attached to the Cu foil tab portion of the negative electrode piece for the working electrode.
• the Ni mesh at the end of the Ni tab for the counter electrode was attached to the corner of the Li piece so as to be perpendicular to the 3.0 cm side of the Li piece for the counter electrode.
• the Ni mesh at the tip of the reference electrode Ni tab was attached to the center of the 1.5 cm side of the Li piece so as to be perpendicular to the 1.5 cm side of the reference electrode Li piece.
• a polypropylene microporous film was sandwiched between the working electrode and the counter electrode, and the reference electrode was close to the working electrode and connected via the polypropylene microporous film so as not to cause a short circuit. This state was sandwiched between two rectangular aluminum-laminated wrapping materials with all the ends of the Ni tabs protruding outside, and the three sides were heat-sealed. Then, an electrolytic solution was injected through the opening. After that, the opening was sealed by heat sealing to prepare a three-electrode laminate type half cell for evaluation. The same electrolytic solution as used in [2-2] above was used.
• This charging/discharging operation was regarded as one cycle, and 20 cycles were performed, and a low rate test was performed in which the above charging/discharging rate was changed to 0.1C in the 21st cycle.
• the discharge capacity at the 50th cycle after the start of the test at 1C was taken as the 50th cycle Li removal capacity.
• the discharge (de-Li) capacity retention rate at the 50th cycle was defined and calculated by the following equation.
• 50th cycle discharge (de-Li) capacity retention rate (%) ⁇ (Capacity for Li removal at 50th cycle after starting 1C test)/(Capacity for removing Li at 1st cycle after starting 1C test) ⁇ ⁇ 100
• Example 1 to 9 As the composite particles, the composite particles (S) and the composite particles (L) are uniformly mixed using a mortar so as to have the composition of the composite particles shown in Table 1 to obtain composite particles 1 to 9. Obtained. The above evaluation was performed on Composite Particles 1 to 9. Table 1 shows the evaluation results.
• the composite particles that satisfy the requirements of the configuration example [1] of one embodiment of the present invention that is, the number of composite particles measured in cross-sectional SEM-EDS, the small particle size composite
• the ratio of the number of particles is 5 to 45%
• Example 9 From Example 9, it can be seen that by coating the composite particles with carbon, the capacity retention rate in the cycle test in the three-electrode laminate type half-cell is improved compared to the case without coating. This is because the coating can lower the resistance.
• the composite particles of Comparative Examples 1 to 3 are composite particles that do not satisfy the requirements of the configuration example [1] of one embodiment of the present invention.
• the silicon content is about 40% by mass as in Comparative Example 1, deterioration of the electrode due to expansion and contraction of the Si—C composite appears to be small, and a high capacity retention rate is obtained.
• the compacted density is relatively low at 1.15 g/cm 3 , and it is expected that only electrodes with low volumetric energy densities will be obtained in practical batteries.
• Comparative Example 3 the ratio of the small-diameter composite particles exceeds 5%, but it can be seen that the effects of the present invention cannot be obtained simply by satisfying this requirement.
• Comparative Examples 1 to 3 in Si—C composites produced without blending composite particles with different particle sizes and silicon contents, the value of (y)/(x) is around 1.00. only slightly fluctuates and does not fall below 0.90.
• the ratio of the small-diameter composite particles is 5% or more and 45% or less and the ratio (y)/(x) is 0.90 or less. I know there is. In such a case, even if the silicon content of the composite particles as a whole is 45% by mass or more, the composite particles can provide a battery with good cycle characteristics. It can be seen from the comparison of ⁇ 3.

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